MCTS for non-deterministic games with very high branching factor for chance nodes
You can try using an "Open-Loop" MCTS approach, instead of the standard "closed-loop" one, and eliminate chance nodes altogether. See, for example, Open Loop Search for General Video Game Playing.

In a "standard" (closed-loop) implementation, you would store a game state in every normal (non-chance) node. Whenever there is a chance event, you would stochastically traverse to one of its children, and then have a normal node with a "deterministic" game state again.

In an open-loop approach, you do not store game states in any node (except possibly the root nodes), because nodes no longer deterministically correspond to specific game states. Every node in an open-loop MCTS approach only corresponds to the sequence of actions that leads to it from the root node. This completely eliminates the need for chance nodes, and results in a significantly smaller tree because you only need a single path in your tree for every possible unique sequence of actions. A single sequence of actions may, depending on stochastic events, lead to a distribution over possible game states.

In every separate MCTS iteration, you would re-generate game states again by applying moves "along the edges" as you traverse through the tree. You also "roll the dice" again for any stochastic events. If your MCTS iteration traverses a certain path of the tree often enough, it will still be able to observe all the possible stochastic events through sampling.

Note that, given an infinite amount of time, the closed-loop approach with explicit chance nodes will likely perform much better. But when you have a small amount of time (as is the case in the real-time video game setting considered in the paper I linked above), an open-loop approach without explicit chance nodes may perform better.



Alternatively, if you prefer the closed-loop approach with explicit chance nodes, you could try some mix of:


Allowing MCTS to prioritise promising parts of the search tree over parts that have not been visited at all (i.e. do not automatically prioritise nodes with $0$ visits). For example, instead of giving unvisited node a value estimate of $\infty$ (this is how you could interpret the automatic selection of them), you could give them a value estimate equal to the value estimate of the parent node, and just apply the UCB1 equation directly.
Use AMAF value estimates / RAVE / GRAVE in your selection phase. This allows you to very quickly learn some crude value estimates for moves that you have never selected in the Selection phase yet, by generalising from observations of playing them in the Play-out phase. I have noticed that the "standard" implementation of RAVE / GRAVE, without an explicit UCB-like exploration term, does not mix well with my previous suggestion of using a non-infinite value estimate for unvisited children. It may be good to consider a UCB-like variant with an explicit exploration term instead.